Second-order-optimal filters on lie groups, in:

Abstract-We provide an explicit formula for the secondorder-optimal nonlinear filter for state estimation of systems on general Lie groups with disturbed measurements of inputs and outputs. Optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). We show that the resulting filter will depend on the choice of affine connection, thus encoding the nonlinear geometry of the state space. For the case of attitude estimation, where we are given a second order (dynamic) mechanical system on the tangent bundle of the special orthogonal group SO(3), and where we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a matrix Riccati differential equation that updates the filter gain

Geometric methods for designing optimal filters on Lie groups

In control theory, the problem of having available good measurements is of primary importance in order to perform good tracking and control. Unfortunately, in real-life applications, sensing systems do not provide direct measurements about the pose (and its rate) of mechanical systems, while, in other situations, measurements are so noisy that require pre-processing to filter out disturbances and biases. These problems could be faced by using filters and observers. In this thesis, we apply a second-order optimal minimum-energy filter constructed on Lie groups to several planar bodies. We start by studying the application of the filter to the matrix Lie group TSE(2), i.e. the tangent bundle of the Special Euclidean group SE(2); moreover, a comparison with the extended Kalman filter is presented. After that, we consider the Chaplygin sleigh case, that is a mechanical system with a nonholonomic constraint. Then, we move our attention to the case of an articulated convoy with hooking constraints. Finally, we apply the filter to a real case scenario consisting of a scaled model representing a parking truck semi-trailer system. Particular attention is posed to the description of the geometric structure that underlies the dynamics and to the choice of the measurement equation, the affine connection, and the other parameters that define the filters. Simulations show the effectiveness of the proposed filters. The use of Lie groups theory for designing the filters is challenging, but the accuracy of the results, obtained considering the geometric structure and the symmetries of the system justifies the effort

Nonlinear Attitude Filtering: A Comparison Study

This paper contains a concise comparison of a number of nonlinear attitude filtering methods that have attracted attention in the robotics and aviation literature. With the help of previously published surveys and comparison studies, the vast literature on the subject is narrowed down to a small pool of competitive attitude filters. Amongst these filters is a second-order optimal minimum-energy filter recently proposed by the authors. Easily comparable discretized unit quaternion implementations of the selected filters are provided. We conduct a simulation study and compare the transient behaviour and asymptotic convergence of these filters in two scenarios with different initialization and measurement errors inspired by applications in unmanned aerial robotics and space flight. The second-order optimal minimum-energy filter is shown to have the best performance of all filters, including the industry standard multiplicative extended Kalman filter (MEKF)

Estimation for bilinear stochastic systems

Three techniques for the solution of bilinear estimation problems are presented. First, finite dimensional optimal nonlinear estimators are presented for certain bilinear systems evolving on solvable and nilpotent lie groups. Then the use of harmonic analysis for estimation problems evolving on spheres and other compact manifolds is investigated. Finally, an approximate estimation technique utilizing cumulants is discussed

Achievable DoF-delay trade-offs for the K-user MIMO interference channel with delayed CSIT

©2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.The degrees of freedom (DoFs) of the K-user multiple-input multiple-output (MIMO) interference channel are studied when perfect, but delayed channel state information is available at the transmitter side (delayed CSIT). Recent works have proposed schemes improving the DoF knowledge of the interference channel, but at the cost of developing transmission involving many channel uses (long delay), thus increasing the complexity at both transmitter and receiver side. This paper proposes three linear precoding strategies, limited to at most three phases, based on the concept of interference alignment, and built upon three main ingredients: delayed CSIT precoding, user scheduling, and redundancy transmission. In this respect, the interference alignment is realized by exploiting delayed CSIT to align the interference at the non-intended receivers along the space-time domain. Moreover, a new framework is proposed where the number of transmitted symbols and duration of the phases is obtained as the solution of a maximization problem, and enabling the introduction of complexity constraints, which allows deriving the achievable DoF as a function of the transmission delay, i.e., the achievable DoF-delay trade-off. Finally, the latter part of this paper settles that the assumption of time-varying channels common along all the literature on delayed CSIT is indeed unnecessary.Peer ReviewedPostprint (author's final draft